His scholarly and unobtrusive demeanour as a young lecturer won the admiration of his colleagues and pupils in St Andrews where his teaching left a permanent mark. While he was essentially a geometer he had considerable interests in other sciences, and it is noteworthy that the classes which he chose to attend in his fourth year of study had been in Anatomy and Chemistry. Crystallography in particular appealed to him; and doubtless these possible outlets influenced his geometrical concepts and led Sommerville to ponder over space filling figures, and gave an early impetus to thoughts in a field which he made peculiarly his own. He had an original mind, and beneath his outward shyness considerable talents lay concealed his intellectual grasp of geometry was balanced by a deftness in making models, and on the aesthetic side by an undoubted talent with the brush. In the course of years he produced a pleasing collection of water colour sketches of New Zealand scenery.
His mathematical work falls naturally into two parts: that of the teacher and that of the original investigator. His text-books which have appeared at regular intervals are a valuable link between the old and the new era in the teaching of geometry at College. They are the Elements of Non-Euclidean Geometry (1914), Analytical Conics (1924), Introduction to the Geometry of n Dimensions (1929), and the recent Three Dimensional Geometry (1934) the appearance of which he did not live to see. All are characterized by a variety of algebraic treatment and a wealth of illustrations and examples, but nowhere, does technical manipulation outrun the geometry. The first of these, a provocative little book, appeared at a time when metrical systems alternative to that of Euclid were known only to the few. It is not surprising that such a teacher carried throughout his life the esteem and appreciation of his students. One of his most distinguished pupils, A C Aitken, writes of the critical time in his own student days when the University of Otago was temporarily without a Professor of Mathematics, and how willingly Sommerville filled the gap by weekly correspondence. The written solutions and comments went far beyond what was necessary for mere elucidation.
Beginning in 1905 Sommerville wrote over thirty original papers and notes which have been published in well known journals at home and abroad. The first, entitled Networks of the Plane in Absolute Geometry (Proc. Roy. Soc. Edinburgh, 25) is typical of the sequel. The main theme is that of combinatory geometry, exemplified by a systematic investigation of The Division of Space by Congruent Triangles and Tetrahedra (1923) in the same journal, and extended to n dimensions (Palermo, 48 (1924), 9-22). Out of this grew the work upon the relations connecting angle sum and the volume of a polytope in space of n dimensions (Proc. Roy. Soc. London (1927)).
In his geometrical interests Sommerville exhibited an unusual independence. Although be was conversant with the more fashionable developments of the subject his own researches are for the most part concerned with the two themes of Non-Euclidean Geometry (in the restricted sense of geometry with a projective metric) and the enumerative and other properties of configurations possessing some degree of regularity or completeness, both themes being extended to n dimensions. His familiarity with Non-Euclidean Geometry must have been almost unique: he treated it as worthy of a detailed study comparable to that accorded to Euclidean Geometry. There may be mentioned as examples of his researches the classification of all types of Non-Euclidean Geometry (including those usually excluded as bizarre); the extension, involving the measurement of generalized angles in higher space, of Euler's Theorem on Polyhedra; space-filling figures; the classification of polytopes (i.e. the generalization, in higher space, of polyhedra): it is typical that this includes polytopes in Non-Euclidean Space. His text-book Introduction to Geometry of n Dimensions gives some notion of his researches in these two directions.
His wide knowledge of other branches of geometry (and incidentally of European languages) are seen most clearly in his Bibliography of Non-Euclidean Geometry, whose title, bereft of its subtitles, is misleading, any work on higher space being included.
Sommerville was ever ready to apply his special gifts to unusual examples, as in his analysis of preferential voting by means of a figure in higher space, and in a highly original analysis of the musical scale. His was a life of unsparing activity, and the fruits of his work will abide. There has passed from Scotland one who had already become her leading geometer of the present century.
For cooperating in this appreciation of the geometrical work I have to thank Dr G Timms.
Published papers of D M Y Sommerville.
Networks of the Plane in Absolute Geometry. Proc. Roy. Soc. Edin., 25 (1905), 392-394.
Semi-Regular Networks of the Plane in Absolute Geometry. Trans. Roy. Soc. Edin., 41 (1906), 725-747.
On the Distribution of the Proper Fractions. Proc. Roy. Soc. Edin., 26 (1906), 116-129.
On the Classification of Frequency- Ratios. Biometrika, 5 (1906), 179-181.
On Links and Knots in Euclidean Space of n Dimensions. Messenger of Math. (2), 36 (1906), 139-144.
On Certain Periodic Properties of Cyclic Compositions of Numbers. Proc. Lond. Math. Soc. (2), 7 (1909), 263-313.
The Pedal Line of the Triangle in Non-Euclidean Geometry. Proc. 5th Intern. Math. Congr., 2 (1912), 93-101.
Description of a Projection-Model of the 600-Cell in Space of Four Dimensions. Proc. Roy. Soc. Edin., 34 (1914), 253-258.
The "Slip-Curves" of an Amsler Planimeter. Phil. Mag. (6), 38 (1919), 489-492.
Quadratic Systems of Circles in Non-Euclidean Geometry. Bull. Amer. Math. Soc., 25 (1919), 161-173.
The Singularities of the Algebraic Trochoids. Proc. Lond. Math. Soc. (2), 18 (1920), 385-392).
Division of Space by Congruent Triangles and Tetrahedra. Proc. Roy. Soc. Edin., 43 (1923), 85-116.
The Regular Divisions of Space of n Dimensions and their Metrical Constants. Rendic. Palermo, 48 (1924), 9-22.
The Relations connecting the Angle-Sums and Volume of a Polytope in Space of n Dimensions. Proc. Roy. Soc. Lond. A, 115 (1927), 103-119.
An Analysis of Preferential Voting. Proc. Roy. Soc. Edin., 48 (1928), 140-160.
Certain Hyperspatial Partitionings connected with Preferential Voting. Proc. Lond. Math. Soc. (2), 28 (1928), 368-382.
Sets of Points Self-Conjugate with respect to a Quadric in n Dimensions. Proc. Edin. Math. Soc. (2) 3 (1932), 6-9.
Metrical Co-ordinates in Non-Euclidean Geometry. Proc. Edin. Math. Soc. (2), 3 (1932). 16-25.
On the Relations between the Rotation-Groups of the Regular Polytopes and Permutation-Groups. Proc. Lond. Math. Soc. (2), 35 (1933), 101-115.
Isohedral and Isogonal Generalizations of the Regular Polyhedra. Proc. Roy. Soc. Edin., 52 (1932), 251-263.
Bibliography of Non-Euclidean Geometry. London, 1911. xii+403.
The Elements of Non-Euclidean Geometry. London, 1914 and 1919. xvi+ 274.
Analytical Conics. London, 1924. vii+310.
An Introduction to the Geometry of n Dimensions. London, 1929. xviii+196.
Analytical Geometry of Three Dimensions. Cambridge, 1934. xvi+416.
Papers by D M Y Sommerville in the Proceedings of the Edinburgh Mathematical Society (First Series).
1. On the number of Independent Conditions involved in the Vanishing of a Rectangular Array, 24, pp. 2-6 (1905).
2. On Certain Projective Configurations in Space of n Dimensions and a Related Problem in Arrangements, 25, pp. 80-90 (1906).
3. A Problem in Voting, 28, pp. 23-24 (1909).
4. Classification of Geometries with Projective Metric, 28, pp. 25-41 (1909).
5. Elementary Considerations relating to the Absolute, 28, pp. 65-72 (1909).
6. Note on the Geometries in which Straight Lines are represented by Circles, 28, pp. 81-94 (1909).
7. Note on Legendre's and Bertrand's Proofs of the Parallel Postulate by Infinite Areas, 30, pp. 31-36 (1911).
8. Note on Gauss's Proof of the Reciprocity of Parallelism, 32, pp. 15-18 (1913).
9. Taylor's Cubics associated with a Triangle in Non-Euclidean Geometry, 33, pp. 85-99 (1914).
10. Approximations to the length of an Arc, 37, 76-79 (1918).
11. Space-filling Tetrahedra in Euclidean Space, 41, pp. 49-57 (1922).
12. Peaucellier's Cell and other Linkages in Non-Euclidean Geometry, 44, pp. 26-31 (1925).
Also three papers in the EMS Mathematical Notes.